<i>Turbulence: The Legacy of A. N. Kolmogorov</i>

Frisch, U.; Donnelly, Russell J.

doi:10.1063/1.881555

Cited by 994 publications

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“…However, very few analytic statements have been derived from these equations [1,2]. In this Letter we consider decaying isotropic turbulence, i.e.…”

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confidence: 99%

“…In studies of the decay of homogeneous and isotropic turbulence self-similarity in the shape of the energy spectra are often assumed, expressed as the principle of permanence of large eddies (PLE) [2]. In a recent interesting phenomenological study it was proposed that for initial spectra not as steep as k 4 there will be three ranges of self-similarity where PLE only applies to the very large scales [4].…”

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confidence: 99%

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Scaling and the prediction of energy spectra in decaying hydrodynamic turbulence

Ditlevsen¹,

Jensen²,

Olesen³

2004

Physica A: Statistical Mechanics and its Applications

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Few rigorous results are derived for fully developed turbulence. By applying the scaling properties of the Navier-Stokes equation we have derived a relation for the energy spectrum valid for unforced or decaying isotropic turbulence. We find the existence of a scaling function ψ. The energy spectrum can at any time by a suitable rescaling be mapped onto this function. This indicates that the initial (primordial) energy spectrum is in principle retained in the energy spectrum observed at any later time, and the principle of permanence of large eddies is derived. The result can be seen as a restoration of the determinism of the Navier-Stokes equation in the mean. We compare our results with a windtunnel experiment and find good agreement.The Navier-Stokes equation (NSE) for hydrodynamic flow has been known for many years, and several interesting numerical results have been found. However, very few analytic statements have been derived from these equations [1,2]. In this Letter we consider decaying isotropic turbulence, i.e. hydrodynamics without external forcing.One well known feature of the hydrodynamic equations is their invariance under re-scaling when the coordinates are scaled by an arbitrary quantity l. Any solution to the NSE can then be mapped onto another solution with a corresponding change in the velocity, the time and the diffusion coefficient ν. The same scaling argument applies to the energy density. In the following we shall show that this leads to a general scaling behavior of the energy density considered as a function of k, t, ν.The result resembles the k 4 backscattering at integral scales obtained in EDQNM closure calculations [3]. In studies of the decay of homogeneous and isotropic turbulence self-similarity in the shape of the energy spectra are often assumed, expressed as the principle of permanence of large eddies (PLE) [2]. In a recent interesting phenomenological study it was proposed that for initial spectra not as steep as k 4 there will be three ranges of self-similarity where PLE only applies to the very large scales [4].We shall now derive the scaling properties of the energy density directly from the scaling properties of the NSE. This has been done by one of the authors in the case where viscosity was ignored [5]. Here we shall generalize these results. Consider the energy density in D dimensions, given bywhere L and K are infrared and ultraviolet cutoffs, respectively, and Ω D is the solid angle. We assume isotropy such that the energy density only depends on the modulus k = |k| of the wave vector. The total energy density is given byUsing the invariance of the unforced NSE under the scalingwhere h is an arbitrary parameter, the relationtrivially follows.In the following we assume the cutoffs such that for the relevant physical range we have 1/L ≪ k ≪ K. We will therefore suppress the explicit dependence of the energy density on the cutoffs. The viscosity provides a physical cutoff for large k while the cutoff 1/L is determined by the boundary conditions at the integral scale.Let...

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“…However, very few analytic statements have been derived from these equations [1,2]. In this Letter we consider decaying isotropic turbulence, i.e.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Scaling and the prediction of energy spectra in decaying hydrodynamic turbulence

Ditlevsen¹,

Jensen²,

Olesen³

2004

Physica A: Statistical Mechanics and its Applications

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“…Structural function (SF) represents an effective tool in theoretical and experimental investigations of chaotic systems and processes, for example, in turbulence investigations (Frisch, 1995 (Vstovsky, 2006). That paper demonstrated effectiveness of such an approach in comparison with many parameter fitness (approximation) of SF.…”

Section: Sfcam Basicsmentioning

confidence: 99%

Factual Revelation of Temporal and Spatial Hierarchical Correlations by Structural Function Curvature Analysis Method

Vstovsky

2012

ESR

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Basics of structural function curvature analysis method (SFCAM) are described shortly and three examples of SFCAM application are described: for revelation of earthquake predictors, surface relief analysis and surface relief evolution description. Discussion is carried out in terms of correlation times (CT) or correlation lengths (CL), respectively to the type of experimental data -time series or surface reliefs. CTs or CLs can be taken into account in two ways: by an order of their recognition, from the shortest to the greatest, or by separation over previously determined classes due to physical sense of investigated structure. On the whole, SFCAM represent a subtle enough tool for analysis of complex hierarchical systems.

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“…This analysis method has been used in various domains, each presenting self-similar data, such as turbulence in physics [1], network traffic [2], DNA series in biology [3] or in the study of natural images [4]. The development of new instruments in several domains, as astrophysics [5] or vision [6], leads to a new generation of data: spherical data.…”

Section: Introductionmentioning

confidence: 99%

Multifractal Analysis on the Sphere

Koenig

Chainais

2008

Lecture Notes in Computer Science

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Abstract.A new generation of instruments in astrophysics or vision now provide spherical data. These spherical data may present a selfsimilarity property while no spherical analysis tool is yet available to characterize this property. In this paper we present a first numerical study of the extension of multifractal analysis onto the sphere using spherical wavelet transforms. We use a model of multifractal spherical textures as a reference to test this approach. The results of the spherical analysis appear qualitatively satisfactory but not as accurate as those of the usual 2D multifractal analysis.

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Turbulence: The Legacy of A. N. Kolmogorov

Cited by 994 publications

References 0 publications

Scaling and the prediction of energy spectra in decaying hydrodynamic turbulence

Scaling and the prediction of energy spectra in decaying hydrodynamic turbulence

Factual Revelation of Temporal and Spatial Hierarchical Correlations by Structural Function Curvature Analysis Method

Multifractal Analysis on the Sphere

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